Calculating Trace of 4x4 Matrix: (A+I)^{-1}

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SUMMARY

The discussion focuses on calculating the trace of the matrix (A+I)^{-1}, where matrix A is defined as a 4x4 matrix with specific values. The result of the trace calculation is confirmed to be 38/15. The participant inquires about the potential benefits of diagonalization in simplifying the calculation process. It is established that diagonalization can indeed facilitate finding the trace of A more efficiently.

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Ahmes
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Hello,
I have to calculate the trace of the following matrix: [tex](A+I)^{-1}[/tex]
where [tex]A[/tex] is:
[tex]1 0 0 1[/tex]
[tex]0 2 2 0[/tex]
[tex]0 2 2 0[/tex]
[tex]1 0 0 1[/tex]
and [tex]I[/tex] is the unit matrix 4x4.
The calculations are extremely excruciating and the result is [tex]38/15[/tex] [checked]. I'm afraid I miss the whole point of this. Can the fact the matrix is diagonalizable help me reach the answer faster?
I mean not going through the process of adding 1 to the diagonal elements and do the inverse.

Thank you.
 
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Can the fact the matrix is diagonalizable help me reach the answer faster?
It would if you could get the trace of A from its diagonalization!
 

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